pushforward

Pushforward refers to the operation of transporting mathematical data along a map from a source space to a target space, producing an object on the target. It is often called the direct image and is conceptually dual to the pullback, which moves data in the opposite direction.

In measure theory and probability, the pushforward is defined for a measurable map f: X → Y and a measure μ on X. The pushforward measure f_* μ on Y is given by f_* μ(A) = μ(f^{-1}(A)) for measurable sets A. This construction preserves total mass and satisfies the relation ∫ φ d(f_* μ) = ∫ φ ∘ f dμ for integrable functions φ. It is compatible with composition: (g ∘ f)_* μ = g_* (f_* μ).

In differential geometry, the pushforward (often called the tangent map) of a smooth map f: M → N between manifolds is its differential df_p: T_p M → T_{f(p)} N at each p ∈ M. This linear map describes how tangent vectors are sent forward along f. Consequently, a vector field X on M can be pushed forward to a vector field on f(M) by p ↦ df_p(X_p), when defined. The pushforward extends to tensors via the differential’s action on indices.

In other areas, pushforward appears as the direct image in algebraic geometry and topology, where it refers to the induced maps on sheaves, homology, or cohomology along a map f. These uses share the common theme of transferring structure along a map to the target space.

Overall, pushforward provides a framework for aggregating or transmitting data from the source to the target under a given mapping, with the specific construction depending on the mathematical context.